for all \((\zeta ,\xi ) \in \mathcal {D}_\sigma \) .
The existence and uniqueness of \((\phi ,\psi )\in \mathcal {D}_\sigma \)  solving (REF ) follows directly from the Lax-Milgram theorem. Next, using regularity theory for elliptic problems with bulk-surface coupling (see [1]}), where \((f-\phi ,g-\psi )\in \mathcal {L}^2\)  is interpreted as the source term,
we infer that \((\phi ,\psi ) \in \mathcal {H}^2\)  and () holds almost everywhere. This entails the desired conclusion.
